Remarks about the thermostatistical description of the HMF model Part II: Phenomenology of Relaxation Dynamics

TL;DR

This paper investigates the relaxation dynamics of the Hamiltonian Mean Field model, proposing a phenomenological Fokker-Planck approach to describe collisional relaxation timescales and their dependence on system parameters.

Contribution

It introduces a new phenomenological Fokker-Planck model based on parametric resonance to explain relaxation timescales in the HMF model.

Findings

Relaxation timescale scales linearly with system size N.

Effective dynamical coexistence explains long relaxation times.

Proposed model supports a specific dependence of relaxation time on system parameters.

Abstract

After a general overview of some features of the relaxation dynamics of the Hamiltonian Mean Field model, its equilibrium thermodynamic properties are used to rephrase the out-of-equilibrium regime for energies below the critical point $u_{c}=0.75$ in terms of an effective dynamical coexistence between a clustered and a gaseous phases, whose existence could be associated to the large relaxation times observed when $u_{1}<u<u_{c}$, with $u_{1}=0.5$. Starting from the hypothesis that the \textit{parametric resonance} is the microscopic mechanism allowing the energetic interchange between the particles during the collisional regime, a phenomenological Fokker-Planck equation based on a Langevin equation with a multiplicative noise is proposed in order to describe the collisional relaxation of this system towards its final equilibrium, which supports the following dependence of the collisional relaxation timescale $\tau_{cr}=\tau_{0}N\equiv\sqrt{IN/g}$.